Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 021, 10 pages      cond-mat/0602427      http://dx.doi.org/10.3842/SIGMA.2006.021

On the Degenerate Multiplicity of the sl2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Tetsuo Deguchi
Department of Physics, Faculty of Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-Ku, Tokyo 112-8610, Japan

Received October 31, 2005, in final form February 06, 2006; Published online February 17, 2006

Abstract
We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl2 loop algebra symmetry if the q parameter is given by a root of unity, q02N = 1, for an integer N. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight dk±, which leads to evaluation parameters aj. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

Key words: loop algebra; the six-vertex model; roots of unity representations of quantum groups; Drinfeld polynomial.

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